Polynomial-Time Key Recovery Attack on the Faure-Loidreau Scheme based on Gabidulin Codes

TL;DR

This paper demonstrates a polynomial-time attack on the Faure-Loidreau rank-metric encryption scheme, revealing vulnerabilities similar to those in McEliece-based systems and compromising its previously claimed security.

Contribution

The paper introduces the first practical polynomial-time key recovery attack on the Faure-Loidreau scheme, exploiting Overbeck's attack on an associated public code.

Findings

Successfully breaks 80-bit security parameters in seconds

Shows the scheme's vulnerability to known attacks on Gabidulin code-based cryptosystems

Highlights the need for revised security assessments of rank-metric schemes

Abstract

Encryption schemes based on the rank metric lead to small public key sizes of order of few thousands bytes which represents a very attractive feature compared to Hamming metric-based encryption schemes where public key sizes are of order of hundreds of thousands bytes even with additional structures like the cyclicity. The main tool for building public key encryption schemes in rank metric is the McEliece encryption setting used with the family of Gabidulin codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and Tretjakov, many systems have been proposed based on different masking techniques for Gabidulin codes. Nevertheless, over the years all these systems were attacked essentially by the use of an attack proposed by Overbeck. In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was not in the McEliece setting. The scheme is very efficient, with small public keys of size a few kiloBytes and with security closely related to the linearized polynomial reconstruction problem which corresponds to the decoding problem of Gabidulin codes. The structure of the scheme differs considerably from the classical McEliece setting and until our work, the scheme had never been attacked. We show in this article that this scheme like other schemes based on Gabidulin codes, is also vulnerable to a polynomial-time attack that recovers the private key by applying Overbeck's attack on an appropriate public code. As an example we break concrete proposed $80$ bits security parameters in a few seconds.